Geometric Sequence Calculator

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Need to find the nth term or sum of a geometric sequence? This geometric sequence calculator computes any term, the sum of n terms, and (when applicable) the sum to infinity.

Geometric Sequence Calculator

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Using the calculator

Enter three values:

First term (a₁) – the starting value of your sequence
Common ratio (r) – the constant factor each term is multiplied by
Number of terms (n) – how many terms to consider

The calculator instantly shows the nth term (aₙ), the sum of n terms (Sₙ), and optionally the full sequence. When |r| < 1, it also shows the sum to infinity.

What is a geometric sequence?

A geometric sequence (or geometric progression) is a sequence where each term is found by multiplying the previous term by a constant called the common ratio.

Examples:

2, 6, 18, 54, 162, ... (first term = 2, ratio = 3)
100, 50, 25, 12.5, ... (first term = 100, ratio = 0.5)
1, -2, 4, -8, 16, ... (first term = 1, ratio = -2)

The nth term formula

To find any term in a geometric sequence directly:

a_n = a_1 \cdot r^{n-1}

Where:

aₙ = the nth term
a₁ = first term
r = common ratio
n = term position

Example: finding the 8th term

For the sequence 3, 6, 12, 24, ... (a₁ = 3, r = 2), find the 8th term:

a_8 = 3 \times 2^{8-1} = 3 \times 128 = 384

The sum formula

To find the sum of the first n terms (for r ≠ 1):

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}

When r = 1, the sequence is constant, so Sₙ = n × a₁.

Example: sum of first 10 terms

For the sequence starting at 1 with r = 2:

S_{10} = 1 \times \frac{1 - 2^{10}}{1 - 2} = \frac{1 - 1024}{-1} = 1023

Sum to infinity

When the common ratio satisfies |r| < 1, the terms shrink toward zero and the sum converges:

S_\infty = \frac{a_1}{1 - r}

For example, with a₁ = 4 and r = 0.5:

S_\infty = \frac{4}{1 - 0.5} = \frac{4}{0.5} = 8

The sequence 4, 2, 1, 0.5, 0.25, ... sums to exactly 8 if continued forever. The calculator shows this when applicable.

Geometric vs arithmetic sequences

Don't confuse geometric and arithmetic sequences:

Arithmetic: Add the same amount each time (2, 5, 8, 11...)
Geometric: Multiply by the same ratio each time (2, 6, 18, 54...)

Arithmetic sequences grow linearly; geometric sequences grow (or decay) exponentially.

Common applications

Geometric sequences appear throughout finance and science:

Compound interest: Investment values grow by a fixed percentage (ratio) – see our Compound Interest Calculator
Investment growth: Project future portfolio values with our Investment Calculator or Future Value Calculator
Radioactive decay: Half-life processes (r = 0.5) – try our Half-Life Calculator
Growth rates: Calculate annual returns with our CAGR Calculator

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PKPK started DQYDJ in 2009 to research and discuss finance and investing and help answer financial questions. He's expanded DQYDJ to build visualizations, calculators, and interactive tools. 
 
 PK lives in New Hampshire with his wife, kids, and dog.

Geometric Sequence Calculator